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\begin{document}
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\title{D-Cliques: Topology can compensate NonIIDness in Decentralized Federated Learning}
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\titlerunning{D-Cliques}
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\author{Aur\'elien Bellet\inst{1}\thanks{Authors in alphabetical order of last names, see Section 'Credits' for respective contributions.} \and
Anne-Marie Kermarrec\inst{2} \and
Erick Lavoie\inst{2}}
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\authorrunning{A. Bellet, A-M. Kermarrec, E. Lavoie}
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\institute{Inria, Lille, France\\
\email{aurelien.bellet@inria.fr} \and
EPFL, Lausanne, Switzerland \\
\email{\{anne-marie.kermarrec,erick.lavoie\}@epfl.ch}\\
}
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\maketitle              % typeset the header of the contribution
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\begin{abstract}
The convergence speed of machine learning models trained with Federated Learning is significantly affected by non-independent and identically distributed (non-IID) data partitions, even more so in a fully decentralized setting without a central server. In this paper, we show that the impact can be significantly reduced by carefully designing the communication topology. We present D-Cliques, a novel topology that reduces gradient bias by grouping nodes in cliques such that their local joint distribution is representative of the global class distribution. We show how D-Cliques can be used to successfully implement momentum, which is critical to efficiently train deep convolutional networks but otherwise detrimental in a non-IID setting. We then present an extensive experimental study on MNIST and CIFAR10 and demonstrate that D-Cliques provides similar convergence speed as a fully-connected topology with a significant reduction in the number of required edges and messages. In a 1000-node topology, D-Cliques requires 98\% less edges and 96\% less total messages to achieve a similar accuracy. Our study suggests that maintaining full connectivity within cliques is necessary for fast convergence, but we show that we can reduce inter-clique connectivity by relying on a small-world topology with a logarithmic number of edges without damaging the convergence speed. D-cliques with a small-world topology leads to a further 22\% reduction in the number of edges in a 1000-node network, with even bigger gains expected at larger scales.

\keywords{Decentralized Learning \and Federated Learning \and Topology \and
Non-IID Data \and Stochastic Gradient Descent}
\end{abstract}
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\section{Introduction}

% 1/ Decentralized FL approaches can be more scalable than Centralized FL approach when the number of nodes is large
% 2/ It is well known the topology can affect convergence of decentralized algorithms, as shown by classic convergence analysis. However the effect of topology has been observed to be often quite small in practice. This is because most of these results were obtained for iid data.
% 3/ In this paper, we show that the effect of topology is very significant for non-iid data. Unlike for centralized FL approaches, this happens even when nodes perform a single local update before averaging. We propose an approach to design a sparse data-aware topology which recovers the convergence speed of a centralized approach.
% 4/ An originality of our approach is to work at the topology level without changing the original efficient and simple D-SGD algorithm. Other work to mitigate the effect of non-iid on decentralized algorithms are based on performing modified updates (eg with variance reduction) or multiple averaging steps.

Machine learning is currently shifting from the classic \emph{centralized}
paradigm, in which models are trained on data located on a single machine or
in a data center, to more \emph{decentralized} ones.
Indeed, data is often inherently decentralized as it is collected by several
parties (such as different hospitals, companies, personal devices...).
Federated Learning (FL) allows a set
of data owners to collaboratively train machine learning models
on their joint
data while keeping it decentralized, thereby avoiding the costs of moving
data as well as mitigating privacy and confidentiality concerns 
\cite{kairouz2019advances}.
Due to the decentralized nature of data collection, the local datasets of
participants can be very different in size and distribution: they are 
\emph{not} independent and identically distributed
(non-IID). In particular, the class distributions may vary a lot
across local datasets \cite{quagmire}.
Therefore, one of the key challenges in FL is to design algorithms that
can efficiently deal with such non-IID data 
\cite{kairouz2019advances,fedprox,scaffold,quagmire}.

Federated learning algorithms can be classified into two categories depending
on the network topology they work on. In server-based FL, the network is
organized as a star: a central server orchestrates the training process and
iteratively aggregates model updates received from the participants
(\emph{clients}) and sends
them back the aggregated model \cite{mcmahan2016communication}. In contrast,
fully decentralized FL algorithms operate over an arbitrary topology where
participants communicate in a peer-to-peer fashion with their direct neighbors
in the network graph. A classic example of such algorithms is Decentralized
SGD (D-SGD) \cite{lian2017d-psgd}, in which participants alternate between
local SGD updates and model averaging with neighboring nodes.

In this work, we focus on fully decentralized algorithms as they can
generally
scale better to the large number of participants seen in ``cross-device''
applications \cite{kairouz2019advances}. Indeed, while a central
server quickly becomes a bottleneck as the number of participants increases, the topology used in fully decentralized algorithms can remain sparse
enough such that all participants have small (constant or logarithmic) degree 
\cite{lian2017d-psgd}. Recent work has shown both empirically 
\cite{lian2017d-psgd,Lian2018} and theoretically \cite{neglia2020} that sparse
topologies like rings or grids do not significantly affect the convergence
rate compared to using denser topologies when data is IID.
% We also note that full decentralization can also provide benefits in terms of
% privacy protection \cite{amp_dec}.

In contrast to the IID case, we show in this work that \emph{the impact of
topology is very significant for non-IID data}. This phenomenon is illustrated
in Figure~\ref{fig:iid-vs-non-iid-problem}, where we see that using a ring or
grid topology completely jeopardize the convergence rate in practice when
classes are imbalanced across participants.
We stress that the fact unlike for centralized FL approaches 
\cite{kairouz2019advances,scaffold,quagmire}, this
happens even when nodes perform a single local update before averaging the
mode with their neighbors. We thus study the following question:

% \textit{Are there regular topologies, i.e. where all nodes have similar or the same number of neighbours, with less connections than a fully-connected graph that retain a similar convergence speed and non-IID behaviour?}

\textit{Are there sparse topologies that allow a similar convergence
speed as the fully connected graph under a large number of participants with
non-IID class distribution?}

We answer this question in the affirmative by proposing \textsc{D-Cliques}, an approach to
design \emph{a sparse data-aware topology which allows to recover the convergence
speed of a centralized (or IID) approach}. Our proposal includes a simple
modification of the standard D-SGD algorithm which ensures that gradients are
unbiased with respect to the class distribution.
We empirically evaluate our approach on MNIST and CIFAR10 datasets using
logistic
regression and deep convolutional models with up to 1000 participants. This is
in contrast to most previous work on fully decentralized algorithms
considering only a few tens of participants \cite{tang18a,more_refs}, which
fall short of
giving a realistic view of the performance of these algorithms in actual
applications.
\aurelien{TODO: complete above paragraph with more details and highlighting
other contributions as needed}

To summarize, our contributions are as follows:
\begin{enumerate}
  \item we show the significant impact of topology on convergence speed in the presence of non-IID data in decentralized learning;
  \item we propose the D-Cliques topology to remove the impact of non-IID data on convergence speed, similar to a fully-connected topology. At a scale of 1000 nodes, this represents a 98\% reduction in the number of edges ($18.9$ vs $999$ edges per node on average) and a 96\% reduction in the total number of required messages;
  \item we show how to leverage cliques to: (1) remove gradient bias that originate from inter-clique edges; 
   (2) implement momentum, a critical optimization technique to quickly train convolutional networks, that otherwise significantly \textit{decreases} convergence speed in the presence of non-IID data;
  \item we show that, among the many possible choices of inter-clique topologies, a smallworld topology provides a convergence speed close to fully-connecting all cliques pairwise, but requires only $O(n + log(n))$ instead of $O(n^2)$ edges where $n$ is the number of nodes. At a scale of 1000 nodes, this represents a further 22\% reduction in the number of edges compared to fully-connecting cliques ($14.6$ vs $18.9$ edges per node on average) and suggests possible bigger gains at larger scales. 
\end{enumerate}

The rest of this paper is organized as follows. \aurelien{TO COMPLETE}

\begin{figure}
     \centering
     \begin{subfigure}[b]{0.34\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/ring-IID-vs-non-IID}
\caption{\label{fig:ring-IID-vs-non-IID} Ring: (almost) minimal connectivity.}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.27\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/grid-IID-vs-non-IID}
\caption{\label{fig:grid-IID-vs-non-IID} Grid: intermediate connectivity.}
     \end{subfigure}
     \hfill
     \begin{subfigure}[b]{0.34\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/fully-connected-IID-vs-non-IID}
\caption{\label{fig:fully-connected-IID-vs-non-IID} Fully-connected: maximal connectivity.}
     \end{subfigure}
        \caption{IID vs non-IID Convergence Speed. Thin lines are the minimum
        and maximum accuracy of individual nodes. Bold lines are the average
        accuracy across all nodes.\protect \footnotemark \aurelien{TODO: make
        the figure more self-contained (adding all information to
        understand the setting) and perhaps add a fig to show the
        effect of D-Cliques}}
        \label{fig:iid-vs-non-iid-problem}
\end{figure}

\footnotetext{This is different from the accuracy of the average model across nodes that is sometimes used once training is completed.}

\section{Problem Statement}

\label{section:problem}

A set of $n$ nodes $N = \{1, \dots, n \}$ communicates with their neighbours defined by the mixing matrix $W$ in which $W_{ij}$ defines the \textit{weight} of the outgoing connection from node $i$ to $j$. $W_{ij} = 0$ means that there is no connection from node $i$ to $j$ and $W_{ij} > 0$ means there is a connection.

Training data is sampled from a global distribution $D$ unknown to the nodes. Each node has access to an arbitrary partition of the samples that follows its own local distribution $D_i$. Nodes cooperate to reach consensus on a global model $M$ that performs well on $D$ by minimizing the average training loss on local models:

\begin{equation}
min_{x_i, i = 1, \dots, n} = \frac{1}{n}\sum_{i=1}^{n} \mathds{E}_{s_i \sim D_i} F_i(x_i;s_i) 
\label{eq:dist-optimization-problem}
\end{equation}

such that $M= x_i = x_j, \forall i,j \in N$, where $x_i$ are the parameters of
node $i$'s local model, $s_i$ is a sample of $D_i$, $F_i$ is the loss function
on node $i$, and $\mathds{E}_{s_i \sim D_i} F_i(x_i;s_i)$ denotes  the
expected value of $F_i$ on a random sample $s_i$ drawn from $D_i$.

\subsection{Non-IID Data}
\label{section:non-iid-assumptions}

Removing the assumption of \textit{independent and identically distributed} (IID) data opens a wide range of potential practical difficulties. While non-IID simply means that a local dataset is a biased sample of the global distribution $D$, the difficulty of the learning problem depends on additional factors that compound with that bias. For example, an imbalance in the number of examples for each class represented in the global distribution compounds with the position of the nodes that have the examples of the rarest class. Additionally, if two local datasets have different number of examples, the examples in the smaller dataset will be visited more often than those in a larger dataset, potentially skewing the optimisation process to perform better on the examples seen more often.

To focus our study while still retaining the core aspects of the problem, we make the following assumptions: (1) all classes are equally represented in the global dataset, by randomly removing examples from the larger classes if necessary; (2) all classes are represented on the same number of nodes; (3) all nodes have the same number of examples. Within those assumptions, we take the hardest possible problem, which is to have each node having examples of only a single class. For the following experiments, we use the MNIST (CITE) and CIFAR10 (CITE) datasets.

\subsection{Learning Algorithm}

We use the Decentralized-Parallel Stochastic Gradient Descent, aka D-PSGD~\cite{lian2017d-psgd}, illustrated in Algorithm~\ref{Algorithm:D-PSGD}. A single step consists of sampling the local distribution $D_i$, computing and applying a stochastic gradient descent (SGD) step with regard to that sample, and averaging the model with its neighbours. Both outgoing and incoming weights of $W$ must sum to 1, i.e. $W$  is doubly stochastic ($\sum_{j \in N} W_{ij} = 1$ and $\sum_{j \in N} W_{ji} = 1$), and communication is symmetric, i.e. $W_{ij} = W_{ji}$.

\begin{algorithm}[h]
   \caption{D-PSGD, Node $i$}
   \label{Algorithm:D-PSGD}
   \begin{algorithmic}[1]
        \State \textbf{Require} initial model parameters $x_i^{(0)}$, learning rate $\gamma$, mixing weights $W$, number of steps $K$, loss function $F$
        \For{$k = 1,\ldots, K$}
          \State $s_i^{(k)} \gets \textit{sample from~} D_i$
          \State $x_i^{(k-\frac{1}{2})} \gets x_i^{(k-1)} - \gamma \nabla F(x_i^{(k-1)}; s_i^{(k)})$ 
          \State $x_i^{(k)} \gets \sum_{j \in N} W_{ji}^{(k)} x_j^{(k-\frac{1}{2})}$
        \EndFor
   \end{algorithmic}
\end{algorithm}

D-PSGD can be used with a variety of models, including deep learning networks. In the rest of this paper, we use it with a linear (regression) model on MNIST, and with a deep convolutional network on CIFAR10.

%To remove the impact of particular architectural choices on our results, we use a linear classifier (CITE). This model provides up to 92.5\% accuracy when fully converged on MNIST (CITE), about 7\% less than state-of-the-art deep learning networks (CITE).

%\subsection{Clustering}
%
%From the perspective of one node, \textit{clustering} intuitively represents how many connections exist between its immediate neighbours. A high level of clustering means that neighbours have many edges between each other. The highest level is a \textit{clique}, where all nodes in the neighbourhood are connected to one another. Formally, the level of clustering, between $0$ and $1$, is the ratio of $\frac{\textit{nb edges between neighbours}}{\textit{nb possible edges}}$~\cite{watts2000small}.
%

\section{D-Cliques: Creating Locally Representative Cliques}
\label{section:d-cliques}

To have a preliminary intuition of the impact of non-IID data on convergence speed, examine the local neighbourhood of a single node in a grid similar to that used to obtain results in Figure~\ref{fig:grid-IID-vs-non-IID}. As illustrated in Figure~\ref{fig:grid-iid-vs-non-iid-neighbourhood}, the color of a node, represented as a circle, corresponds to one of the 10 available classes in the dataset. In this IID setting (Figure~\ref{fig:grid-iid-neighbourhood}), each node has examples of all ten classes in equal proportions. In  the other (rather extreme) non-IID case (Figure~\ref{fig:grid-non-iid-neighbourhood}), each node has examples of only a single class and nodes are distributed randomly in the grid: this particular example illustrates a neighbourhood with two nodes with examples of the same class adjacent to each other.

\begin{figure}
     \centering
     \begin{subfigure}[b]{0.25\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/grid-iid-neighbourhood}
\caption{\label{fig:grid-iid-neighbourhood} IID}
     \end{subfigure}
     \begin{subfigure}[b]{0.25\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/grid-non-iid-neighbourhood}
\caption{\label{fig:grid-non-iid-neighbourhood}  Non-IID}
     \end{subfigure}
        \caption{Neighbourhood in an IID and non-IID Grid.}
        \label{fig:grid-iid-vs-non-iid-neighbourhood}
\end{figure}

For the sake of the argument, assume all nodes are initialized with the same model weights, which is not critical for quick convergence but makes the comparison easier. A single training step, from the point of view of the middle node of Figure~\ref{fig:grid-IID-vs-non-IID}, is equivalent to sampling a mini-batch five times larger from the union of the local distributions of the five illustrated nodes. 

In the IID case, since gradients are computed from examples of all classes, the resulting average gradient will point in a direction that lowers the loss for all classes. This is the case because the components of the gradient that would only improve the loss on a subset of the classes to the detriment of others are cancelled by similar but opposite components from other classes. Therefore only the components that improve the loss for all classes remain. 

However, in the (rather extreme) non-IID case illustrated, there are not enough nodes in the neighbourhood to remove the bias of the classes represented. Even if all nodes start from the same model weights, they will diverge from one another according to the classes represented in their neighbourhood, more than they would have had in the IID case. Moreover, as the distributed averaging algorithm takes several steps to converge, this variance is never fully resolved and the variance remains between steps.\footnote{It is possible, but impractical, to compensate for this effect by averaging multiple times before the next gradient computation. In effect, this trades connectivity (number of edges) for latency to give the same convergence speed, in number of gradients computed, as a fully connected graph.} This additional variance biases subsequent gradient computations as the gradients are computed further away from the global average, in addition to being computed from different examples. As shown in Figure~\ref{fig:ring-IID-vs-non-IID} and \ref{fig:grid-IID-vs-non-IID}, this significantly slows down convergence speed to the point of making parallel optimization impractical.

\begin{algorithm}[h]
   \caption{D-Clique Construction}
   \label{Algorithm:D-Clique-Construction}
   \begin{algorithmic}[1]
        \State \textbf{Require} set of classes globally present $L$, 
        \State~~ set of all nodes $N = \{ 1, 2, \dots, n \}$,
        \State~~ fn $\textit{classes}(S)$ that returns the classes present in a subset of nodes $S$,
        \State~~ fn $\textit{intraconnect}(DC)$ that returns edges intraconnecting cliques of $DC$,
        \State~~ fn $\textit{interconnect}(DC)$ that returns edges interconnecting cliques of $DC$ (Sec.~\ref{section:interclique-topologies})
         \State~~ fn $\textit{weights}(E)$ that assigns weights to edges in $E$ 
        \State $R \leftarrow \{ n~\text{for}~n \in N \}$ \Comment{Remaining nodes}
        \State $DC \leftarrow \emptyset$ \Comment{D-Cliques}
        \State $\textit{C} \leftarrow \emptyset$ \Comment{Current Clique}
        \While{$R \neq \emptyset$}
		\State $n \leftarrow \text{pick}~1~\text{from}~\{ m \in R | \textit{classes}(\{m\}) \subsetneq \textit{classes}(\textit{C}) \}$
		\State $R \leftarrow R \setminus \{ n \}; C \leftarrow C \cup \{ n \}$
		\If{$\textit{classes}(C) = L$}
		    \State $DC \leftarrow DC \cup \{ C \}; C \leftarrow \emptyset$
		\EndIf
        \EndWhile
        \State \Return $weights(\textit{intraconnect}(DC) \cup \textit{interconnect}(DC))$
   \end{algorithmic}
\end{algorithm}

Under our non-IID assumptions (Section~\ref{section:non-iid-assumptions}), a balanced representation of classes, similar to that of the IID case, can be recovered by modifying the topology such that each node has direct neighbours of all classes. Moreover, as we shall show in the next sections, there are benefits in ensuring the clustering of neighbours into a \textit{clique}, such that, within a clique, neighbours of a node are also directly connected. To ensure all cliques still converge to a single model, a number of inter-clique connections are introduced, established directly between nodes that are part of cliques. Because the joint location distributions $D_{\textit{clique}} = \sum_{i \in \textit{clique}} D_i$ is representative of the global distribution, similar to the IID case, a sparse topology can be used between cliques, significantly reducing the total number of edges required to obtain quick convergence. And because the number of connections required per node is low and even, this approach is well suited to decentralized federated learning. 

The construction of the resulting \textit{decentralized cliques} (d-cliques) topology can be performed with Algorithm~\ref{Algorithm:D-Clique-Construction}. Essentially, each clique $C$ are constructed one at a time by selecting nodes with differing classes. Once all cliques are constructed, intra-clique and inter-clique edges are added. 

Finally, weights are assigned to edges to ensure quick convergence, for this study we use Metropolis-Hasting (CITE), which while not offering optimal convergence speed in the general case, provides good convergence by taking into account the degree of immediate neighbours:

\begin{equation}
  W_{ij} = \begin{cases}
    max(\text{degree}(i), \text{degree}(j)) + 1 & \text{if}~i \neq j \\
   1 - \sum_{j \neq i} W_{ij} & \text{otherwise}
  \end{cases}
\end{equation}

In this paper, we focus on showing the convergence benefits of such a topology for decentralized federated learning. Algorithm~\ref{Algorithm:D-Clique-Construction} therefore centrally generates the topology, which is then tested in a simulator. We expect this algorithm should be straightforward to adapt for a decentralized execution: the computation of the classes globally present, $L$, could be computed PushSum (CITE), and the section of neighbours done with PeerSampling (CITE).

\begin{figure}[htbp]
    \centering 
             
    \begin{subfigure}[b]{0.4\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/fully-connected-cliques}
    \caption{\label{fig:d-cliques-figure} D-Cliques Connected Pairwise}
    \end{subfigure}
    \hfill
    % To regenerate figure, from results/mnist
    % python ../../../learn-topology/tools/plot_convergence.py fully-connected/all/2021-03-10-09:25:19-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:49-CET --add-min-max --yaxis test-accuracy --ymin 80 --ymax 92.5 --labels '100 nodes non-IID fully-connected' '100 nodes non-IID d-cliques' --save-figure ../../figures/d-cliques-mnist-vs-fully-connected.png --legend 'lower right' --font-size 16
    \begin{subfigure}[b]{0.55\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/d-cliques-mnist-vs-fully-connected.png}
    \caption{\label{fig:d-cliques-example-convergence-speed} Convergence Speed on MNIST. Y-axis starts at 80.}
    \end{subfigure}
    
\caption{\label{fig:d-cliques-example} D-Cliques}
\end{figure}

Using Algorithm~\ref{Algorithm:D-Clique-Construction} on a network of 100 nodes generates the topology illustrated in Figure~\ref{fig:d-cliques-figure}, with the convergence speed of Figure~\ref{fig:d-cliques-example-convergence-speed}. The convergence speed is quite close to that of a fully-connected topology, and significantly better than that of the ring and grid of Figure~\ref{fig:iid-vs-non-iid-problem}. At a scale of 100 nodes, it uses only $\approx10\%$ of the number of edges of a fully-connected topology, offering a reduction of $\approx90\%$. Nonetheless, there is still significant variance in accuracy between nodes, which we address in the next section by removing the bias actually introduced by inter-clique edges.



%The degree of \textit{skew} of local distributions $D_i$, i.e. how much the local distribution deviates from the global distribution on each node, influences the minimal size of cliques. 
%
%The global distribution of classes, for classification tasks, can be computed from the distribution of class examples on the nodes, with Distributed Averaging (CITE). Given the global distribution of classes, neighbours within cliques can be chosen based on a PeerSampling (CITE) service. Both services can be implemented such that they converge in a logarithmic number of steps compared to the number of nodes. It is therefore possible to obtain this information in a scalable way.
%
% In the rest of this paper, we assume these services are available and show that the approach provides a useful convergence speed after the cliques have been formed.



\section{Removing Gradient Bias from Inter-Clique Edges}
\label{section:clique-averaging}

Inter-clique connections create sources of bias. By averaging models after a gradient step, D-PSGD effectively gives a different weight to the gradient of neighbours. 

\begin{figure}[htbp]
         \centering
         \includegraphics[width=0.5\textwidth]{figures/connected-cliques-bias}
\caption{\label{fig:connected-cliques-bias} Sources of Bias in Connected Cliques: Non-uniform weights in neighbours of A (A has a higher weight); Non-uniform class representation in neighbours of B (extra green node).}
\end{figure}

Figure~\ref{fig:connected-cliques-bias} illustrates the problem with the simplest case of two cliques connected by one inter-clique edge connecting the green node of the left clique with the purple node of the right clique.
Node A will have a weight of $\frac{12}{110}$ while all of A's neighbours will have a weight of $\frac{11}{110}$, except the green node connected to B, that will have a weight of $\frac{10}{110}$. This weight assignment therefore biases the gradient towards A's class and aways from the green class. The same analysis holds for all other nodes without inter-clique edges. For node B, all neighbours and B will have weights of $\frac{1}{11}$. However, the green class is represented twice while all other classes are represented only once. This biases the gradient toward the green class. The combined effect of these two sources of bias is to increase the variance between models after a D-PSGD step of training.

\begin{algorithm}[h]
   \caption{Clique-Unbiased D-PSGD, Node $i$}
   \label{Algorithm:Clique-Unbiased-D-PSGD}
   \begin{algorithmic}[1]
        \State \textbf{Require} initial model parameters $x_i^{(0)}$, learning rate $\gamma$, mixing weights $W$, number of steps $K$, loss function $F$
        \For{$k = 1,\ldots, K$}
          \State $s_i^{(k)} \gets \textit{sample from~} D_i$
          \State $g_i^{(k)} \gets \frac{1}{|\textit{Clique}(i)|}\sum_{j \in \textit{Clique(i)}}  \nabla F(x_j^{(k-1)}; s_j^{(k)})$
          \State $x_i^{(k-\frac{1}{2})} \gets x_i^{(k-1)} - \gamma g_i^{(k)}$ 
          \State $x_i^{(k)} \gets \sum_{j \in N} W_{ji}^{(k)} x_j^{(k-\frac{1}{2})}$
        \EndFor
   \end{algorithmic}
\end{algorithm}

We solve this problem with a clique-unbiased version of D-PSGD, listed in Algorithm~\ref{Algorithm:Clique-Unbiased-D-PSGD}: gradient averaging is decoupled from weight averaging by sending each in separate rounds of messages. Only the gradients of neighbours within the same clique are used to compute the average gradient, which provides an equal representation to all classes in the computation of the average gradient. But the models of all neighbours, including those across inter-clique edges, are used for computing the distributed average of models as in the original version.

% To regenerate figure, from results/mnist:
% python ../../../learn-topology/tools/plot_convergence.py fully-connected/all/2021-03-10-09:25:19-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:49-CET  no-init/fully-connected-cliques/all/2021-03-12-11:12:01-CET --add-min-max --yaxis test-accuracy --labels '100 nodes non-IID fully-connected' '100 nodes non-IID d-cliques w/o clique avg.' '100 nodes non-IID w/ clique avg.' --legend 'lower right' --ymin 89 --ymax 92.5 --font-size 13 --save-figure ../../figures/d-clique-mnist-clique-avg.png
\begin{figure}[htbp]
         \centering
         \includegraphics[width=0.55\textwidth]{figures/d-clique-mnist-clique-avg}
\caption{\label{fig:d-clique-mnist-clique-avg} Effect of Clique Averaging on MNIST. Y-axis starts at 89.}
\end{figure}

As illustrated in Figure~\ref{fig:d-clique-mnist-clique-avg}, this significantly reduces variance between nodes and accelerates convergence speed: the node with lowest accuracy performs as well as the average nodes when not using clique averaging. The convergence speed is now essentially identical to that obtained when fully connecting all nodes. These benefits are obtained at a higher messaging cost, double to that without clique averaging, and increases latency of a single training step by requiring two rounds of messages. Nonetheless, compared to fully connecting all nodes, the total number of messages is reduced by $\approx 80\%$. MNIST and a Linear model are relatively simple, so the next section shows to work with a harder dataset and a higher capacity model.

\section{Implementing Momentum with Clique Averaging}

Training higher capacity models, such as a deep convolutional network, on harder datasets, such as CIFAR10, is usually done with additional optimization techniques to accelerate convergence speed in centralized settings. But sometimes, these techniques rely on an IID assumption in local distributions which does not hold in more general cases. We show here how Clique Averaging (Section~\ref{section:clique-averaging}) easily enables the implementation of these optimization techniques in the more general non-IID setting with D-Cliques.

In particular, we implement momentum (CITE), which increases the magnitude of the components of the gradient that are shared between several consecutive steps. Momentum is critical for making deep convolutional networks, such as LeNet, converge quickly. However, a simpler application of momentum in a non-IID setting can actually be detrimental. As illustrated in Figure~\ref{fig:d-cliques-cifar10-momentum-non-iid-effect}, the convergence of LeNet on CIFAR10 with momentum with 100 nodes using the d-cliques topology is so bad that the network actually fails to converge.  To put things in context, we compare the convergence speed to a single centralized IID node performing the same number of updates per epoch, therefore using a batch size 100 larger: this is essentially equivalent to completely removing the impact of the topology, non-IIDness, and decentralized averaging on the convergence speed. As shown, not using momentum gives a better convergence speed, but this is still far off from the one that would be obtained with a single centralized IID node, so momentum is actually necessary.

\begin{figure}[htbp]
    \centering 
    % To regenerate figure, from results/cifar10
    % python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-13:52:58-CET  no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET no-init-no-clique-avg-no-momentum/fully-connected-cliques/all/2021-03-26-13:47:35-CET/ --legend 'upper right' --add-min-max --labels '1-node IID w/ momentum'  '100 nodes non-IID d-cliques w/ momentum' '100 nodes non-IID d-cliques w/o momentum'  --font-size 14 --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-momentum-non-iid-effect.png --ymax 100         
    \begin{subfigure}[b]{0.45\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-momentum-non-iid-effect}
    \caption{\label{fig:d-cliques-cifar10-momentum-non-iid-effect} Without Clique Averaging }
    \end{subfigure}
    \hfill
    % To regenerate figure, from results/cifar10
    % python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-13:52:58-CET no-init/fully-connected-cliques/all/2021-03-13-18:32:55-CET --legend 'upper right' --add-min-max --labels '1-node IID w/ momentum' '100 nodes non-IID d-clique w/ momentum' --font-size 14 --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-momentum-non-iid-clique-avg-effect.png --ymax 100
    \begin{subfigure}[b]{0.45\textwidth}
    \centering
    \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-momentum-non-iid-clique-avg-effect}
    \caption{\label{fig:d-cliques-cifar10-momentum-non-iid-clique-avg-effect} With Clique Averaging}
    \end{subfigure}
\caption{\label{fig:cifar10-momentum} Non-IID Effect of Momentum on CIFAR10 with LeNet}
\end{figure}

Using D-Cliques (Section~\ref{section:d-cliques}) and Clique Averaging (Section~\ref{section:clique-averaging}), unbiased momentum can be calculated from the clique-unbiased average gradient $g_i^{(k)}$ of Algorithm~\ref{Algorithm:Clique-Unbiased-D-PSGD}:
\begin{equation}
v_i^{(k)} \leftarrow m v_i^{(k-1)} +  g_i^{(k)} 
\end{equation}
It then suffices to modify the original gradient step to use momentum:
\begin{equation}
x_i^{(k-\frac{1}{2})} \leftarrow x_i^{(k-1)} - \gamma v_i^{(k)} 
\end{equation}

Using momentum closes the gap, with a slightly lower convergence speed in the first 20 epochs, as illustrated in Figure~\ref{fig:d-cliques-cifar10-momentum-non-iid-clique-avg-effect}. We expect a similar approach could also enable other optimization techniques (CITE) to be usable in non-IID settings.

 \section{Comparison to Similar Non-Clustered Topologies}

We have previously shown that D-Cliques, can effectively provide similar convergence speed as a fully-connected topology and even a single IID node. We now show, in this section and the next, that the particular structure of D-Cliques is necessary. In particular, we show that similar results may not necessarily be obtained from a similar number of edges chosen at random. We therefore compare d-cliques, with and without clique averaging, to a random topology chosen such that each node has exactly 10 edges, which is similar and even slightly higher than the 9.9 edges on average of the D-Clique topology of Fig.~\ref{fig:d-cliques-figure} on 100 nodes. To better understand the effect of clustering, we also compare to a similar random topology where edges are chosen such that each node has neighbours of all possible classes but without them forming a clique. We finally also compare with an analogous of Clique Averaging, where all nodes de-bias their gradient with that of their neighbours, but since nodes do not form a clique, no node actually compute the same resulting average gradient.

Results for MNIST and CIFAR10 are shown in Figure~\ref{fig:d-cliques-comparison-to-non-clustered-topologies}. For MNIST, a random topology has higher variance and lower convergence speed than D-Cliques, with or without Clique Averaging. However, a random topology with enforced diversity performs as well and even slightly better than D-Cliques without Clique Averaging. Suprisingly, a random topology with unbiased gradient performs worse  than without, but only marginally, so this does not seem quite significant. Nonetheless, the D-Cliques topology with Clique Averaging performs better than any other random topology so it seems clustering in this case has a small but significant effect.

\begin{figure}[htbp]
     \centering     
         \begin{subfigure}[b]{0.48\textwidth}
% To regenerate the figure, from directory results/mnist
% python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-10:19:44-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:49-CET  random-10/all/2021-03-17-20:28:12-CET  random-10-diverse/all/2021-03-17-20:28:35-CET --labels 'd-clique (fcc)' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes represented)' --add-min-max --legend 'lower right' --ymin 88 --ymax 92.5 --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-linear-comparison-to-non-clustered-topologies.png --font-size 13
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-linear-comparison-to-non-clustered-topologies}
                  \caption{MNIST with Linear Model}
         \end{subfigure}
                 \hfill                      
% To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py no-init/fully-connected-cliques/all/2021-03-13-18:32:55-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET random-10/all/2021-03-17-20:30:03-CET  random-10-diverse/all/2021-03-17-20:30:41-CET random-10-diverse-unbiased-gradient/all/2021-03-17-20:31:14-CET --labels 'd-clique (fcc) clique avg.' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes repr.)' '10 random (all classes repr.) with unbiased grad.' --add-min-max --legend 'upper left' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies.png --ymax 119 --font-size 13
        \begin{subfigure}[b]{0.48\textwidth}
        \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies}
         \caption{CIFAR10 with LeNet}
     \end{subfigure} 
 \caption{\label{fig:d-cliques-comparison-to-non-clustered-topologies} Comparison to Non-Clustered Topologies} 
\end{figure}

For CIFAR10, the result is more dramatic, as Clique Averaging is critical for convergence (with momentum). All random topologies fail to converge, except when combining both node diversity and unbiased gradient, but in any case D-Cliques with Clique Averaging converges significantly faster. This suggests clustering helps reducing variance between nodes and therefore helps with convergence speed. We have tried to use LeNet on MNIST to see if the difference between MNIST and CIFAR10 could be attributed to the capacity difference between the Linear and Convolutional networks, whose optimization may benefit from clustering (see Appendix). The difference is less dramatic than for CIFAR10, so it must be that the dataset also has an impact but the exact nature of it is still an open question.

\section{Importance of Intra-Clique Full Connectivity}

Having established that clustering, i.e. the creation of cliques, has a significant effect, we evaluate the necessity of intra-clique full connectivity. Figure~\ref{fig:d-cliques-intra-connectivity} shows the convergence speed of D-Cliques with respectively 1 and 5 edges randomly removed, out of 45 (2 and 10 out of 90 if counting both direction separately), as well as with and without Clique Averaging (resulting in a biased average gradient within cliques). In all cases, both for MNIST and CIFAR10, it has significant effect on the convergence speed. In the case of CIFAR10, it also negates the benefits of D-Cliques. Full-connectivity within cliques is therefore necessary.

\begin{figure}[htbp]
     \centering

\begin{subfigure}[htbp]{0.48\textwidth}
     \centering   
% To regenerate the figure, from directory results/mnist
% python ../../../learn-topology/tools/plot_convergence.py no-init/fully-connected-cliques/all/2021-03-12-11:12:01-CET rm-1-edge/all/2021-03-18-17:28:27-CET rm-5-edges/all/2021-03-18-17:29:10-CET rm-1-edge-unbiased-grad/all/2021-03-18-17:28:47-CET rm-5-edges-unbiased-grad/all/2021-03-18-17:29:36-CET --add-min-max --ymin 85 --ymax 92.5 --legend 'lower right' --yaxis test-accuracy --labels 'fcc with clique grad.' 'fcc -1 edge/clique, no clique avg.' 'fcc -5 edges/clique, no clique avg.'  'fcc -1 edge/clique, clique avg.' 'fcc -5 edges/clique, clique avg.' --save-figure ../../figures/d-cliques-mnist-clique-clustering-fcc.png  --font-size 13   
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-clique-clustering-fcc}     
\caption{\label{fig:d-cliques-mnist-clique-clustering} MNIST}
\end{subfigure}
\hfill
\begin{subfigure}[htbp]{0.48\textwidth}
     \centering
% To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py no-init/fully-connected-cliques/all/2021-03-13-18:32:55-CET rm-1-edge/all/2021-03-18-17:29:58-CET rm-5-edges/all/2021-03-18-17:30:38-CET rm-1-edge-unbiased-grad/all/2021-03-18-17:30:17-CET rm-5-edges-unbiased-grad/all/2021-03-18-17:31:04-CET --add-min-max --ymax 80 --legend 'upper left' --yaxis test-accuracy --labels 'fcc, clique grad.' 'fcc -1 edge/clique, no clique grad.' 'fcc -5 edges/clique, no clique grad.'  'fcc -1 edge/clique, clique grad.' 'fcc -5 edges/clique, clique grad.' --save-figure ../../figures/d-cliques-cifar10-clique-clustering-fcc.png --font-size 13
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-clique-clustering-fcc}
\caption{\label{fig:d-cliques-cifar10-clique-clustering} CIFAR10}
\end{subfigure}

\caption{\label{fig:d-cliques-intra-connectivity} Importance of Intra-Clique Full-Connectivity}
\end{figure}

\section{Scaling with Different Inter-Clique Topologies}
\label{section:interclique-topologies}

We finally evaluate the effect of the inter-clique topology on convergence speed on a larger network of 1000 nodes, dividing the batch size by 10, so the number of updates per epoch remains constant compared to the previous results for 100 nodes. We compare the scalability and resulting convergence speed of different scheme based around D-Cliques, and therefore all using $O(nc)$ edges to create cliques as a foundation, where $n$ is the number of nodes and $c$ is the size of a clique.

First, the scheme that uses the fewest (almost\footnote{A path uses one less edge at significantly slower convergence speed and is therefore never really used in practice.}) number of extra edges is a \textit{ring}. A ring adds $\frac{n}{c} - 1$ inter-clique edges and therefore scales linearly in $O(n)$.

Second, surprisingly (to us), another scheme also scales linearly with a logarithmic bound on the averaging shortest number of hops between nodes, which we call "\textit{fractal}". In this scheme, when the number of nodes keeps growing, cliques are assembled in larger groups of $c$ cliques that are connected internally with one edge per pair of cliques, but with only one edge between pairs of larger groups. The scheme is recursive such that $c$ groups will themselves form a larger group the next level up. This scheme results in at most $nc$ edges per node if edges are evenly distributed, and therefore also scales linearly in the number of nodes.

Third, cliques may also be connected in a smallworld-like~\cite{watts2000small} topology, that may be reminiscent of distributed-hash table designs such as Chord (CITE). In this scheme, cliques are first arranged in a ring as in the first scheme. Then each clique add symmetrically one edge, both clockwise and counter-clockwise on the ring, to the $k$ closest cliques in sets of cliques that are exponentially bigger the further they are on the ring, as detailed in Algorithm~\ref{Algorithm:Smallworld}. This ensures good clustering with other cliques that are close on the ring, while still keeping the average shortest path small (including nodes further on the ring). This scheme adds a $2klog(\frac{n}{c})$ inter-clique edges and therefore grows in the order of $O(n + log(n))$ with the number of nodes.

\begin{algorithm}[h]
   \caption{$\textit{smallworld}(DC)$:  adds $O(\# N + log(\# N))$ edges}
   \label{Algorithm:Smallworld}
   \begin{algorithmic}[1]
        \State \textbf{Require} Set of cliques $DC$ (set of set of nodes), size of neighbourhood $ns$ (default 2), function $\textit{least\_edges}(S, E)$ that returns one of the nodes in $S$ with the least number of edges in $E$
        \State $E \leftarrow \emptyset$ \Comment{Set of Edges}
        \State $L \leftarrow [ C~\text{for}~C \in DC ]$ \Comment{Arrange cliques in a list}
        \For{$i \in \{1,\dots,\#DC\}$} \Comment{For every clique}
          \State \Comment{For sets of cliques exponentially further away from $i$}
          \For{$\textit{offset} \in \{ 2^x~\text{for}~x~\in \{ 0, \dots, \lceil \textit{log}_2(\#DC) \rceil \} \}$} 
             \State \Comment{Pick the $ns$ closests}
             \For{$k \in \{0,\dots,ns-1\}$}
                 \State \Comment{Add interclique connections in both directions}
                 \State $n \leftarrow \textit{least\_edges}(L_i, E)$
                 \State $m \leftarrow \textit{least\_edges}(L_{(i+\textit{offset}+k) \% \#DC}, E)$ \Comment{clockwise in ring}
                 \State $E \leftarrow E \cup \{ (n,m), (m,n) \}$
                 \State $n \leftarrow \textit{least\_edges}(L_i, E)$
                 \State $m \leftarrow \textit{least\_edges}(L_{(i-\textit{offset}-k)\% \#DC} , E)$ \Comment{counter-clockwise in ring}
                 \State $E \leftarrow E \cup \{ (n,m), (m,n) \}$
             \EndFor
           \EndFor
        \EndFor
        \State \Return E
   \end{algorithmic}
\end{algorithm}

Finally, we can also fully connect cliques together, which bounds the average shortest path to $2$ between any pair of nodes. This adds $\frac{n}{c}(\frac{n}{c} - 1)$ edges, which scales quadratically in the number of nodes in $O(\frac{n^2}{c^2})$, which can be significant at larger scales when $n$ is large compared to $c$.

Figure~\ref{fig:d-cliques-cifar10-convolutional} shows convergence speeds for all schemes, both on MNIST and CIFAR10, compared to a single node IID performing the same number of updates per epoch (showing the faster convergence speed achievable if topology had no impact). A ring converges but is much slower. Our "fractal" scheme helps significantly, while still scaling linearly in the number of nodes. But the sweet spot really seems to be with a smallworld topology, as the convergence speed is almost the same to a fully-connected topology, but uses 22\% less edges at that scale (14.5 edges on average instead of 18.9), and seems to have potential to have larger benefits at larger scales. Nonetheless, even the fully-connected topology offers significant benefits with 1000 nodes, as it represents a 98\% reduction in the number of edges compared to fully connecting individual nodes (18.9 edges on average instead of 999) and a 96\% reduction in the number of messages (37.8 messages per round per node on average instead of 999). 

\begin{figure}[htbp]
     \centering
 % To regenerate the figure, from directory results/mnist
 % python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-09:20:03-CET ../scaling/1000/mnist/fully-connected-cliques/all/2021-03-14-17:56:26-CET ../scaling/1000/mnist/smallworld-logn-cliques/all/2021-03-23-21:45:39-CET ../scaling/1000/mnist/fractal-cliques/all/2021-03-14-17:41:59-CET ../scaling/1000/mnist/clique-ring/all/2021-03-13-18:22:36-CET     --add-min-max --yaxis test-accuracy --legend 'lower right' --ymin 84 --ymax 92.5 --labels '1 node IID'  'd-cliques (fully-connected cliques)' 'd-cliques (smallworld)' 'd-cliques (fractal)' 'd-cliques (ring)'  --save-figure ../../figures/d-cliques-mnist-1000-nodes-comparison.png --font-size 13
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
            \includegraphics[width=\textwidth]{figures/d-cliques-mnist-1000-nodes-comparison}
             \caption{\label{fig:d-cliques-mnist-1000-nodes-comparison} MNIST with Linear}
     \end{subfigure}
     \hfill
     % To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-13:52:58-CET ../scaling/1000/cifar10/fully-connected-cliques/all/2021-03-14-17:41:20-CET ../scaling/1000/cifar10/smallworld-logn-cliques/all/2021-03-23-22:13:57-CET  ../scaling/1000/cifar10/fractal-cliques/all/2021-03-14-17:42:46-CET ../scaling/1000/cifar10/clique-ring/all/2021-03-14-09:55:24-CET  --add-min-max --yaxis test-accuracy --labels '1-node IID' 'd-cliques (fully-connected cliques)' 'd-cliques (smallworld)' 'd-cliques (fractal)' 'd-cliques (ring)' --legend 'lower right' --save-figure ../../figures/d-cliques-cifar10-1000-vs-1-node-test-accuracy.png --font-size 13
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-1000-vs-1-node-test-accuracy}
\caption{\label{fig:d-cliques-cifar10-1000-vs-1-node-test-accuracy}  CIFAR10 with LeNet}
     \end{subfigure}
     
\caption{\label{fig:d-cliques-cifar10-convolutional} D-Cliques Convergence Speed with 1000 nodes, non-IID, Constant Updates per Epoch, with Different Inter-Clique Topologies.}
\end{figure}

\section{Related Work}

\aurelien{not sure yet if it is better to have this section here or earlier,
we'll see}

\aurelien{TODO: where to place TornadoAggregate and related refs?}

\paragraph{Dealing with non-IID data in server-based FL.}
Dealing with non-IID data in server-based FL has
recently attracted a lot of interest. While non-IID data is not an issue if
clients send their parameters to the server after each gradient update,
problems arise when one seeks to reduce
the number of communication rounds by allowing each participant to perform
multiple local updates, as in the popular FedAvg algorithm 
\cite{mcmahan2016communication}. This led to the design of extensions that are
specifically designed to mitigate the impact of non-IID data when performing
multiple local updates, using adaptive sampling \cite{quagmire}, update
corrections \cite{scaffold} or regularization in the local objective 
\cite{fedprox}. Another direction is to embrace the non-IID scenario by
learning personalized models for each client 
\cite{smith2017federated,perso_fl_mean,maml,moreau}.

\paragraph{Dealing with non-IID data in fully decentralized FL.}
Non-IID data is known to negatively impact the convergence speed
of fully decentralized FL algorithms in practice \cite{jelasity}. This
motivated the design of algorithms with modified updates based on variance
reduction \cite{tang18a}, momentum correction \cite{momentum_noniid},
cross-gradient
aggregation \cite{cross_gradient}, or multiple averaging steps
between updates (see \cite{consensus_distance} and references therein). These
algorithms
typically require additional communication and/or computation.\footnote{We
also observed that \cite{tang18a} is subject to numerical
instabilities when run on topologies other than rings and grids. When
the rows and columns of $W$ do not exactly
sum to $1$ (due to finite precision), these small differences get amplified by
the proposed updates and make the algorithm diverge.}\aurelien{emphasize that
they only do small scale experiments}
% non-IID known to be a problem for fully decentralized FL. cf Jelasity paper
% D2 and other recent papers on modifying updates: Quasi-Global Momentum,
% Cross-Gradient Aggregation
% papers using multiple averaging steps
% also our personalized papers
% D2 \cite{tang18a}: numerically unstable when $W_{ij}$ rows and columns do not exactly
% sum to $1$, as the small differences are amplified in a positive feedback loop. More work is therefore required on the algorithm to make it usable with a wider variety of topologies. In comparison, D-cliques do not modify the SGD algorithm and instead simply removes some neighbor contributions that would otherwise bias the direction of the gradient. D-Cliques with D-PSGD are therefore as tolerant to ill-conditioned $W_{ij}$ matrices as regular D-PSGD in an IID setting.
In contrast, D-cliques focuses on the design of a sparse topology which is
able to compensate for the effect of non-IID data. We do not modify the simple
and efficient D-SGD
algorithm \cite{lian2017d-psgd} beyond removing some neighbor
contributions
that would otherwise bias the direction of the gradient.
\aurelien{add personalized models - or merge all that in specific paragraph}

% An originality of our approach is to focus on the effect of topology
% level without significantly changing the original simple and efficient D-SGD
% algorithm \cite{lian2017d-psgd}. Other work to mitigate the effect of non-IID
% data on decentralized algorithms are based on performing modified updates (eg
% with variance reduction) or multiple averaging steps.


\paragraph{Impact of topology in fully decentralized FL.} It is well
known
that the choice of network topology can affect the
convergence of fully decentralized algorithms: this is typically accounted
for in the theoretical convergence rate by a dependence on the spectral gap of
the network, see for instance 
\cite{Duchi2012a,Colin2016a,lian2017d-psgd,Nedic18}.
However, for IID data, practice contradicts these classic
results: fully decentralized algorithms converge essentially as fast
on sparse topologies like rings or grids as they do on a fully connected
graph \cite{lian2017d-psgd,Lian2018}. Recent work 
\cite{neglia2020,consensus_distance} sheds light on this phenomenon with refined convergence analyses based on differences between gradients or parameters across nodes, which are typically
smaller in the IID case. However, these results do not give any clear insight
regarding the role of the topology in the non-IID case. We note that some work
has gone into designing efficient topologies to optimize the use of
network resources (see e.g., \cite{marfoq}), but this is done independently
of how data is distributed across nodes. In summary, the role
of topology in the non-IID data scenario is not well understood and we are not
aware of prior work focusing on this question. Our work shows that an
appropriate choice of data-dependent topology can effectively compensate for
non-IID data.

\section{Conclusion}


%\section{Future Work}
%\begin{itemize}
%  \item Non-uniform Class Representation
%  \item End-to-End Wall-Clock Training Time, including Clique Formation
%  \item Comparison to Shuffling Data in a Data Center
%  \item Behaviour in the Presence of Churn
%  \item Relaxing Clique Connectivity: Randomly choose a subset of clique neighbours to compute average gradient.
%\end{itemize}


\section{Credits}

%
% ---- Bibliography ----
%
% BibTeX users should specify bibliography style 'splncs04'.
% References will then be sorted and formatted in the correct style.
%
 \bibliographystyle{splncs04}
 \bibliography{main}
 
 \newpage
 \appendix
 
 \section{Other Experiments}
 
 % REMOVED: Constant Batch-size
%         % To regenerate the figure, from directory results/scaling
%% python ../../../learn-topology/tools/plot_convergence.py 10/mnist/fully-connected-cliques/all/2021-03-10-14:40:35-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fully-connected-cliques/all/2021-03-10-16:44:35-CET --labels '10 nodes bsz=128' '100 nodes bsz=128' '1000 nodes bsz=128 (45)' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fully-connected-cst-bsz.png --ymin 80 --add-min-max
%         \begin{subfigure}[b]{0.48\textwidth}
%         \centering
%         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-scaling-fully-connected-cst-bsz}
%         \caption{FCC: Constant Batch-Size}
%     \end{subfigure}

    \begin{figure}[htbp]
     \centering
     % To regenerate the figure, from directory results/mnist
     % python ../../../learn-topology/tools/plot_convergence.py clique-ring/all/2021-03-10-18:14:35-CET no-clique-avg/clique-ring/all/2021-03-12-10:40:37-CET no-init/clique-ring/all/2021-03-12-10:40:11-CET no-init-no-clique-avg/clique-ring/all/2021-03-12-10:41:03-CET --add-min-max --yaxis test-accuracy --labels '  'with uniform init., with clique avg.'    'with uniform init., without clique avg.'  'without uniform init., with clique avg.' 'without uniform init., without clique avg.' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy.png   
      \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy}
         \caption{\label{fig:d-cliques-mnist-init-clique-avg-effect-ring-test-accuracy} Ring}
     \end{subfigure}
     % To regenerate the figure, from directory results/mnist
     %python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-10:19:44-CET no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:26-CET no-init/fully-connected-cliques/all/2021-03-12-11:12:01-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-12-11:12:49-CET --add-min-max --yaxis test-accuracy --labels 'with uniform init., with clique avg.'    'with uniform init., without clique avg.'  'without uniform init., with clique avg.' 'without uniform init., without clique avg.' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy.png
       \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy}
         \caption{\label{fig:d-cliques-mnist-init-clique-avg-effect-fcc-test-accuracy} Fully-Connected}
     \end{subfigure}
\caption{\label{fig:d-cliques-mnist-init-clique-avg-effect} MNIST: Effects of Clique Averaging and Uniform Initialization on Convergence Speed. (100 nodes, non-IID, D-Cliques, bsz=128)}
\end{figure}

    \begin{figure}[htbp]
     \centering
     % To regenerate the figure, from directory results/cifar10
     % python ../../../learn-topology/tools/plot_convergence.py clique-ring/all/2021-03-10-11:58:43-CET no-init/clique-ring/all/2021-03-13-18:28:30-CET no-clique-avg/clique-ring/all/2021-03-13-18:27:09-CET  no-init-no-clique-avg/clique-ring/all/2021-03-13-18:29:58-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg., with uniform init.' 'with clique avg., without uniform init.'  'without clique avg., with uniform init.'   'without clique avg., without uniform init.' --legend 'lower right' --ymax 75  --save-figure ../../figures/d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy.png  
      \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy}
         \caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect-ring-test-accuracy} Ring}
     \end{subfigure}
     % To regenerate the figure, from directory results/cifar10
     %python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-13:58:57-CET no-init/fully-connected-cliques/all/2021-03-13-18:32:55-CET no-clique-avg/fully-connected-cliques/all/2021-03-13-18:31:36-CET  no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET --add-min-max --yaxis test-accuracy --labels 'with clique avg., with uniform init.' 'with clique avg., without uniform init.'  'without clique avg., with uniform init.'   'without clique avg., without uniform init.' --legend 'lower right'  --ymax 75 --save-figure ../../figures/d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy.png 
       \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy}
         \caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect-fcc-test-accuracy} Fully-Connected}
     \end{subfigure}
\caption{\label{fig:d-cliques-cifar10-init-clique-avg-effect} CIFAR10: Effects of Clique Averaging and Uniform Initialization on Convergence Speed. (100 nodes, non-IID, D-Cliques, bsz=20)}
\end{figure}

\begin{figure}[htbp]
     \centering
% To regenerate the figure, from directory results/mnist
% python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-09:20:03-CET fully-connected/all/2021-03-10-09:25:19-CET clique-ring/all/2021-03-10-18:14:35-CET fully-connected-cliques/all/2021-03-10-10:19:44-CET --add-min-max --yaxis test-accuracy --labels '1-node IID bsz=12800' '100-nodes non-IID fully-connected bsz=128' '100-nodes non-IID D-Cliques (Ring) bsz=128' '100-nodes non-IID D-Cliques (Fully-Connected) bsz=128' --legend 'lower right' --ymin 85 --ymax 92.5 --save-figure ../../figures/d-cliques-mnist-vs-1-node-test-accuracy.png
         \centering
         \includegraphics[width=0.7\textwidth]{figures/d-cliques-mnist-vs-1-node-test-accuracy}
         \caption{\label{fig:d-cliques-mnist-linear-w-clique-averaging-w-initial-averaging} MNIST: D-Cliques Convergence Speed (100 nodes, Constant Updates per Epoch)}
\end{figure}    
     
 \begin{figure}[htbp]
     \centering
          % To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-13:52:58-CET clique-ring/all/2021-03-10-11:58:43-CET fully-connected-cliques/all/2021-03-10-13:58:57-CET --add-min-max --yaxis training-loss --labels '1-node IID bsz=2000' '100-nodes non-IID D-Cliques (Ring) bsz=20' '100-nodes non-IID D-Cliques (Fully-Connected) bsz=20' --legend 'lower right' --save-figure ../../figures/d-cliques-cifar10-vs-1-node-training-loss.png
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-vs-1-node-training-loss}
\caption{\label{fig:d-cliques-cifar10-training-loss} Training Loss}
     \end{subfigure}
     \hfill
     % To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py 1-node-iid/all/2021-03-10-13:52:58-CET clique-ring/all/2021-03-10-11:58:43-CET fully-connected-cliques/all/2021-03-10-13:58:57-CET --add-min-max --yaxis test-accuracy --labels '1-node IID bsz=2000' '100-nodes non-IID D-Cliques (Ring) bsz=20' '100-nodes non-IID D-Cliques (Fully-Connected) bsz=20' --legend 'lower right' --save-figure ../../figures/d-cliques-cifar10-vs-1-node-test-accuracy.png
     \begin{subfigure}[b]{0.48\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-vs-1-node-test-accuracy}
\caption{\label{fig:d-cliques-cifar10-test-accuracy}  Test Accuracy}
     \end{subfigure}
\caption{\label{fig:d-cliques-cifar10-convolutional} D-Cliques Convergence Speed with Convolutional Network on CIFAR10 (100 nodes, Constant Updates per Epoch).}
\end{figure}

\subsection{Scaling behaviour as the number of nodes increases?}
          
     \begin{figure}[htbp]
         \centering     
              % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py 10/mnist/fully-connected-cliques/all/2021-03-12-09:13:27-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fully-connected-cliques/all/2021-03-14-17:56:26-CET --labels '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fully-connected-cst-updates.png --ymin 80 --add-min-max

      \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-scaling-fully-connected-cst-updates}
         \caption{Fully-Connected (Cliques), $O(\frac{n^2}{c^2} + nc)$ edges}
     \end{subfigure}
     
          % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py 10/mnist/clique-ring/all/2021-03-13-18:22:01-CET ../mnist/fully-connected-cliques/all/2021-03-10-10:19:44-CET 1000/mnist/fractal-cliques/all/2021-03-14-17:41:59-CET --labels '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-fractal-cliques-cst-updates.png --ymin 80 --add-min-max
         \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-scaling-fractal-cliques-cst-updates}
         \caption{Fractal, $O(nc)$ edges}
     \end{subfigure}  

     
     % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py 10/mnist/clique-ring/all/2021-03-13-18:22:01-CET ../mnist/clique-ring/all/2021-03-10-18:14:35-CET 1000/mnist/clique-ring/all/2021-03-13-18:22:36-CET --labels '10 nodes bsz=1280' '100 nodes bsz=128' '1000 nodes bsz=13' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-mnist-scaling-clique-ring-cst-updates.png --ymin 80 --add-min-max
         \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-scaling-clique-ring-cst-updates}
         \caption{Ring, $O(n)$ edges}
     \end{subfigure}  
     
     \caption{\label{fig:d-cliques-mnist-scaling-fully-connected} MNIST: D-Clique Scaling Behaviour, where $n$ is the number of nodes, and $c$ the size of a clique (Constant Updates per Epoch).}
     \end{figure}
     
          \begin{figure}[htbp]
         \centering
     
              % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/fully-connected-cliques/all/2021-03-10-13:58:57-CET 1000/cifar10/fully-connected-cliques/all/2021-03-14-17:41:20-CET --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2' --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-fully-connected-cst-updates.png --add-min-max

      \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-scaling-fully-connected-cst-updates}
         \caption{Fully-Connected (Cliques), $O(\frac{n^2}{c^2} + nc)$ edges}
     \end{subfigure}
     
          % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py  ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/fully-connected-cliques/all/2021-03-10-13:58:57-CET 1000/cifar10/fractal-cliques/all/2021-03-14-17:42:46-CET  --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2'  --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-fractal-cliques-cst-updates.png --add-min-max
         \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-scaling-fractal-cliques-cst-updates}
         \caption{Fractal, $O(nc)$ edges}
     \end{subfigure}  

     
     % To regenerate the figure, from directory results/scaling
% python ../../../learn-topology/tools/plot_convergence.py  ../cifar10/1-node-iid/all/2021-03-10-13:52:58-CET 10/cifar10/fully-connected-cliques/all/2021-03-13-19:06:02-CET ../cifar10/clique-ring/all/2021-03-10-11:58:43-CET 1000/cifar10/clique-ring/all/2021-03-14-09:55:24-CET  --labels '1 node IID bsz=2000' '10 nodes non-IID bsz=200' '100 nodes non-IID bsz=20' '1000 nodes non-IID bsz=2'   --legend 'lower right' --yaxis test-accuracy --save-figure ../../figures/d-cliques-cifar10-scaling-clique-ring-cst-updates.png --add-min-max
         \begin{subfigure}[b]{0.7\textwidth}
         \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-scaling-clique-ring-cst-updates}
         \caption{Ring, $O(n)$ edges}
     \end{subfigure}  
     
     \caption{\label{fig:d-cliques-cifar10-scaling-fully-connected} CIFAR10: D-Clique Scaling Behaviour, where $n$ is the number of nodes, and $c$ the size of a clique (Constant Updates per Epoch).}
     \end{figure}
     
     \begin{figure}
\centering
              \begin{subfigure}[htb]{0.48\textwidth}
% To regenerate the figure, from directory results/mnist/gn-lenet
% python ../../../../learn-topology/tools/plot_convergence.py no-init/all/2021-03-22-21:39:54-CET no-init-no-clique-avg/all/2021-03-22-21:40:16-CET random-10/all/2021-03-22-21:41:06-CET random-10-diverse/all/2021-03-22-21:41:46-CET random-10-diverse-unbiased-grad/all/2021-03-22-21:42:04-CET --legend 'lower right' --add-min-max --labels 'd-clique (fcc) clique avg.' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes repr.)' '10 random edges (all classes repr.) with unbiased grad.' --ymin 80 --yaxis test-accuracy --save-figure ../../../figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies.png
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies}
         \caption{\label{fig:d-cliques-mnist-lenet-comparison-to-non-clustered-topologies} LeNet Model}
        \end{subfigure}
        \hfill
                      \begin{subfigure}[htb]{0.48\textwidth}
% To regenerate the figure, from directory results/mnist/gn-lenet
% python ../../../../learn-topology/tools/plot_convergence.py no-init/all/2021-03-22-21:39:54-CET no-init-no-clique-avg/all/2021-03-22-21:40:16-CET random-10/all/2021-03-22-21:41:06-CET random-10-diverse/all/2021-03-22-21:41:46-CET random-10-diverse-unbiased-grad/all/2021-03-22-21:42:04-CET --legend 'upper right' --add-min-max --labels 'd-clique (fcc) clique avg.' 'd-clique (fcc) no clique avg.' '10 random edges' '10 random edges (all classes repr.)' '10 random edges (all classes repr.) with unbiased grad.' --ymax 0.7 --yaxis scattering --save-figure ../../../figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering.png
         \includegraphics[width=\textwidth]{figures/d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering}
         \caption{\label{fig:d-cliques-mnist-lenet-comparison-to-non-clustered-topologies-scattering} LeNet Model (Scattering)}
        \end{subfigure}
       
         \caption{\label{fig:d-cliques-mnist-comparison-to-non-clustered-topologies} MNIST: Comparison to non-Clustered Topologies}
\end{figure}

 \begin{figure}
 \centering
     % To regenerate the figure, from directory results/cifar10
% python ../../../learn-topology/tools/plot_convergence.py fully-connected-cliques/all/2021-03-10-13:58:57-CET no-init-no-clique-avg/fully-connected-cliques/all/2021-03-13-18:34:35-CET  random-10/all/2021-03-17-20:30:03-CET  random-10-diverse/all/2021-03-17-20:30:41-CET random-10-diverse-unbiased-gradient/all/2021-03-17-20:31:14-CET random-10-diverse-unbiased-gradient-uniform-init/all/2021-03-17-20:31:41-CET --labels 'd-clique (fcc) clique avg., uniform init.' 'd-clique (fcc) no clique avg. no uniform init.' '10 random edges' '10 random edges (all classes repr.)' '10 random (all classes repr.) with unbiased grad.' '10 random (all classes repr.) with unbiased grad., uniform init.' --add-min-max --legend 'upper right' --yaxis scattering --save-figure ../../figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering.png --ymax 0.7
        \begin{subfigure}[b]{0.48\textwidth}
        \centering
         \includegraphics[width=\textwidth]{figures/d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering}
         \caption{\label{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies-scattering} LeNet Model: Scattering}
     \end{subfigure}  
         
\caption{\label{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies} CIFAR10: Comparison to non-Clustered Topologies}
\end{figure} 


\begin{itemize}
  \item Clustering does not seem to make a difference in MNIST, even when using a higher-capacity model (LeNet) instead of a linear model. (Fig.\ref{fig:d-cliques-mnist-comparison-to-non-clustered-topologies})     
  \item Except for the random 10 topology, convergence speed seems to be correlated with scattering in CIFAR-10 with LeNet model (Fig.\ref{fig:d-cliques-cifar10-linear-comparison-to-non-clustered-topologies}). There is also more difference between topologies both in convergence speed and scattering than for MNIST (Fig.~\ref{fig:d-cliques-mnist-comparison-to-non-clustered-topologies}). Scattering computed similar to Consensus Control for Decentralized Deep Learning~\cite{kong2021consensus}.
\end{itemize}
 
\end{document}
